ch7 notes (21oct)

# ch7 notes (21oct) - October 21, 2009 CH. 7 Notes dB/B = rdt...

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October 21, 2009 CH. 7 Notes dB/B = rdt ds/s = μ dt + σ dW Take demand: C = g(S(T)) Take value of demand C @ time t, which is: C(t,s(t)) Ito (this is on the pdf fln7 notes, page 5) dC(t,s(t)) = ( C t + σ 2 S 2 )/2 * C ss + μ *S*C s ) * dt + σ *S*C s dW now make share book X π d X π = [r X π + π ( μ -r)] dt + σ π dW X π (t,S(t)) = C(t,s(t)) Vt (I) σ *S* C s = π *r π = S* C s (II) C t + ( σ 2 S 2 /2) * C ss = r * X π + + π *( μ -r) C t + ( σ 2 S 2 /2) * C ss + μ * s * C s – rC – S* C s ( μ -r) = 0 C t + ( σ 2 S 2 /2) * C ss + r*S*C s – rC = 0

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F-K solution for: C t + ( σ 2 S 2 /2) * C ss + r*S*C s – r*C = 0 C (T, S(T)) = g (S(T)) is C (t,x) = E s(t)=x [e -r(T-t) * g * (S(T))] where: ds/s = r dt + σ dW = μ dt + σ dW * dW * = dW + [( μ -r) / σ ] dt EXAMPLE!! C (o,x) = E * s(0)=x [e -rT * g * (S(T))] = E [e -r T * (S(T)-k)] = E [e -r T S(0)*e r T e ( σ W(T) – ( σ ^2)/2 * T ) - e -r T * k = S(0) * E [e ( σ W(T) – ( σ ^2)/2 * T )] - e -r T * k
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## This note was uploaded on 11/12/2009 for the course ECON hag taught by Professor Gunnar during the Fall '09 term at Uni. Iceland.

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ch7 notes (21oct) - October 21, 2009 CH. 7 Notes dB/B = rdt...

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